Telegram needs a phone number register but then you can text people only by username.
Both Signal and WhatsApp have announced usernames, Signal already has a beta version of the app with username support out.
They (Signal) made this[0] blog post last week with APK downloads for testing it in the staging environment (completely without phone number). So if you want to test it, you can already do that right now.
I think it will hit production in early 2024, pretty soon.
Whatsapp is also going forward with a similar feature as a reaction to the Signal updates [1], although not entirely sure if it will work with no phone number at all.